Joe -- That's true, and that can be useful as long as you are all on the same page with what unit-less quantities you are using. When you perform calculations with unit-less quantities, that simply means that you are normalizing the quantities such that certain useful physical constants (like c or G) have a value of 1. But there are many different conventions for such so-called "Natural" units. In the end, normalizing c to 1 is no different than normalizing the length of the king's forearm to 1. It's just a different style of units.

Incidentally, in science, some would argue that just like a value without units is meaningless, a value without some measure of uncertainty is almost as meaningless. However, this would be an experimentalist's viewpoint, and is not as widely held, so your personal mileage may vary.